Prove the following by using the principle of mathematical induction for all $space space straight n element of straight N.$

$1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space........ space plus space straight n.3 to the power of straight n space equals space fraction numerator left parenthesis 2 straight n minus 1 right parenthesis space 3 to the power of straight n plus 1 end exponent space plus space 3 over denominator 4 end fraction$

Solution

Let P(n): $1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space.......... space plus space straight n.3 to the power of straight n space equals space fraction numerator left parenthesis 2 straight n minus 1 right parenthesis to the power of 3 straight n plus 1 end exponent plus 3 over denominator 4 end fraction$
I. For n = 1,
P(1) : $1.3 space equals space fraction numerator left square bracket 2 left parenthesis 1 right parenthesis minus 1 right square bracket 3 to the power of 1 plus 1 end exponent plus 3 over denominator 4 end fraction space rightwards double arrow space 3 space equals space fraction numerator 1 left parenthesis 3 squared right parenthesis plus 3 over denominator 4 end fraction space rightwards double arrow space 3 space equals space fraction numerator 9 plus 3 over denominator 4 end fraction space rightwards double arrow space 3 space equals space 3$

∴ P(1) is true.
II. Suppose the statement is true for n = m, $straight m element of space straight N$
 P(m): $space space 1.3 space plus space 2.3 squared plus 3.3 cubed plus..... space plus space straight m.3 to the power of straight m space equals space fraction numerator left parenthesis 2 straight m minus 1 right parenthesis 3 to the power of straight m plus 1 end exponent plus 3 over denominator 4 end fraction$ ...(i)
III. For n = (m + 1),
 P(m + 1) : $1.3 space plus space 2.3 squared plus 3.3 cubed space plus space......... space plus space left parenthesis straight m plus 1 right parenthesis space 3 to the power of straight m plus 1 end exponent space equals space fraction numerator open square brackets 2 left parenthesis straight m plus 1 right parenthesis space minus 1 close square brackets 3 to the power of straight m plus 1 plus 1 end exponent plus 3 over denominator 4 end fraction$
or $1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space....... plus space straight m.3 to the power of straight m plus left parenthesis straight m plus 1 right parenthesis 3 to the power of straight m plus 1 end exponent space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis.3 to the power of straight m plus 2 end exponent plus 3 over denominator 4 end fraction$
 From (i),
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∴ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow space space space fraction numerator left parenthesis 2 straight m minus 1 right parenthesis space 3 to the power of straight m plus 1 end exponent over denominator 4 end fraction plus space 3 over 4 space plus space left parenthesis straight m plus 1 right parenthesis space 3 to the power of straight m plus 1 end exponent space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis 3 to the power of straight m plus 2 end exponent space plus space 3 over denominator 4 end fraction$
$rightwards double arrow space space space space space space 3 to the power of straight m plus 1 end exponent open square brackets fraction numerator 2 straight m minus 1 over denominator 4 end fraction plus left parenthesis straight m plus 1 right parenthesis close square brackets space plus space 3 over 4 space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis 3 to the power of straight m plus 2 end exponent plus 3 over denominator 4 end fraction$
$space space rightwards double arrow space space space space 3 to the power of straight m plus 1 end exponent open square brackets fraction numerator 6 straight m plus 3 over denominator 4 end fraction close square brackets space plus space 3 over 4 space equals space fraction numerator left parenthesis 2 straight m space plus space 1 right parenthesis 3 to the power of straight m plus 2 end exponent space plus space 3 over denominator 4 end fraction space rightwards double arrow space fraction numerator 3 to the power of straight m plus 1 end exponent.3 left parenthesis 2 straight m space plus 1 right parenthesis over denominator 4 end fraction space plus 3 over 4 space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis.3 to the power of straight m plus 2 end exponent space plus 3 over denominator 4 end fraction$
$rightwards double arrow space space space space space space space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis to the power of 3 straight m plus 2 end exponent plus 3 over denominator 4 end fraction space equals space fraction numerator left parenthesis 2 straight m plus 1 right parenthesis 3 to the power of straight m plus 2 end exponent plus 3 over denominator 4 end fraction$
Which is true

∴ P(m + 1) is true

∴ P(m) is true $rightwards double arrow$ P(m + 1) is true
Hence, by the principle of mathematical induction, statement P(n) is true for all $straight n space element of space straight N.$

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